A dynamic theory of the corporate finance based upon
repeated signaling∗
Christopher A. Hennessy
Dmitry Livdan
Bruno Miranda
First Draft: April 25, 2006
Current Draft: May 1, 2007
Abstract
We examine the effect of repeated hidden information regarding the marginal product of capital on
the dynamics of financing and investment. The model features endogenous investment, debt, default,
dividends, equity flotations and share repurchases. Since signaling costs are necessarily high if net worth is
low, forward-looking shareholders have a precautionary motive to hoard cash and capital. In each period’s
least-cost separating equilibrium, firms signal positive information with high leverage and overinvest in
order to reduce associated default costs. Firms with negative information save (rather than borrow) and
raise external funds with equity. The need for costly signals is increasing in a firm’s financing gap. If
net worth is sufficiently low, extreme costs of signaling induce a switch to pooling equilibria where firms
with positive (negative) information underinvest (overinvest). The model is rich in testable predictions
and consistent with a broad set of stylized facts regarding leverage ratios and announcement effects.
1
Introduction
Three decades have elapsed since Ross (1977) and Leland and Pyle (1977) introduced the “incentive signaling”
approach to corporate finance. However, we still lack a dynamic (infinite-horizon) quantitative theory of
financing and investment in economies with hidden information.1 This was recently noted by Ross (2005)
himself, who stated, “The introduction of the issues raised by the presence of asymmetric information
in the determination of the capital structure and the integration of these issues into the intertemporal
neoclassical model are a major challenge.” Recently, a number of researchers have tried to analyze the
∗ The authors are from U.C. Berkeley, Texas A&M, and UCLA, respectively. We thank Stephen Ross for encouraging us
to work on this topic as well as seminar participants at Columbia, Dartmouth College, NYU, LBS, LSE, Bank of England,
HEC Lausanne, Minnesota, University of Vienna, Washington University of St. Louis, and the 2006 Lone Star Conference.
We also thank Heitor Almeida, Sudipto Bhattacharya, Patrick Bolton, Dirk Hackbarth, Erwan Morellec, Holger Mueller, Tom
Noe, Michael Roberts, Rajdeep Singh, Alexei Tchistyi, Toni Whited and Andrew Winton for detailed feedback. This paper was
previously circulated under a different title.
1 In contrast, substantial progress has been made on dynamic models with taxes and hidden action. For recent examples,
see Gamba and Triantis (2006) and DeMarzo and Fishman (2007), respectively.
1
effect of hidden information on investment by adopting the intuitively plausible assumption that it gives
rise to linear-quadratic costs of external equity.2 Recognizing the potential problems associated with this
approach, Gomes (2001) states, “Ideally we would prefer to model financial intermediation endogenously.”
In this paper, we start from primitives and develop a dynamic quantitative model of corporate financing and
investment when insiders have private information about the marginal product of capital. We argue that
the proposed theory can compete with structural trade-off theoretic models in terms of explaining a broad
range of stylized facts.
We start with a neoclassical investment framework. The firm has a stochastic concave profit function
θt εt ktα . The shock εt is public and observed simultaneously by all agents at the end of the period. The shock
θt ∈ {θL , θH } is privately observed by an insider at the start of the period, with outsiders observing the
private shock with a one-period lag. Using his private information, the insider maximizes the cum-dividend
value of his equity stake. The insider is forward-looking and recognizes that his decisions affect the severity
of future adverse selection problems.
Anticipation of signaling costs necessitated by repeated hidden information causes risk-neutral insiders
to exhibit pseudo-risk-averse preferences with equity value concave in net worth.3 To see this, suppose the
insider has positive information about the marginal product of capital, but net worth is low. Here the firm
would need to raise a large amount of funds in order to implement first-best investment. However, the
issuance of a large block of equity would create a strong temptation for firms with negative information to
mimic, since they stand to gain a great deal from security mispricing. In order to separate, the insider with
positive information must utilize costly signals such as defaultable debt. For the firm with low net worth,
the marginal value of internal funds is high, since they reduce the need for external funds and concomitant
signaling costs. At the opposite extreme, a firm with high net worth does not require external funding. Such
a firm will use a marginal dollar to pay dividends, implying that internal funds have a shadow value of one.
The least-costly separating equilibrium (LCSE) minimizes the temptation of insiders with negative
information to mimic. Since insiders are pseudo-risk-averse, firms reporting low expected profitability are
financed with external equity, while still holding a cash buffer stock. In this way, outside investors provide the
firm with insurance in the form of financial slack. In contrast, firms can credibly signal positive information
by substituting debt for equity. Using the optimal mix of real and financial signals, investment of the high
type is distorted above first-best, as the firm tries to lower expected default costs by increasing capital. As we
show, similar signaling effects are also present in the static analog of our dynamic model, but precautionary
motives are absent. Thus, in a setting with repeated hidden information, firms hoard more cash, issue less
debt, and invest more in physical capital. The precautionary savings incentives arising from repeated hidden
information offer a potential explanation for the debt conservatism puzzle posited by Graham (2000).
The possibility of pooling equilibria is addressed in an extension of the dynamic model. In order to deter
2 For
examples, see Gomes (2001), Cooley and Quadrini (2001), and Hennessy and Whited (2006).
Mariotti, Plantin and Rochet (2007) and DeMarzo and Fishman (2007) show that repeated hidden action generates
concave continuation values in infinite-horizon models of the firm.
3 Biais,
2
imitation, high types must incur significant signaling costs in order separate when the financing gap is large.
Absent such costly signals, the low type would gain from overvaluation of a large block of securities. As
intuition would suggest, a pooling equilibrium Pareto dominates the LCSE if net worth is sufficiently low. In
the pooling equilibrium, firms with positive (negative) private information underinvest (overinvest) relative
to first-best. In the model, firms pool in low net worth states and separate via costly signals at intermediate
levels of net worth. If net worth is sufficiently high, the external funding requirement is low, and firms with
negative information have no motivation to mimic high types in order to gain from securities mispricing. In
such states, policies resemble the full-information first-best, but saving and investment are somewhat higher
due to endogenous precautionary motives.
A technical contribution of the paper is to develop a tractable algorithm to solve for equilibrium in a
dynamic neoclassical setting with hidden information. This is important inasmuch as analogous symmetric
information models are central in macroeconomics (e.g. Lucas and Prescott (1971)), production-based asset
pricing (e.g. Cochrane (1991)), and corporate finance (e.g. Gomes (2001)). The solution method is logically
the same as that used in static signaling models, e.g. Milgrom and Roberts (1986). The first additional
source of complexity is that budget and incentive constraints involve endogenous value functions, while
single-period models involve terminal cash flows. The second additional source of complexity is determining
the endogenous default threshold accounting for the option value inherent in continuation. The two hurdles
are readily cleared using recursive methods (e.g. Stokey and Lucas (1989)). Following Maskin and Tirole
(1992) and Tirole (2006), a Pareto optimality criterion allows us to determine the net worth levels at which
pooling is possible and when the LCSE is a unique equilibrium. Again, recursive methods must be exploited
since changes in equilibria lead to changes in value functions, and vice versa.
The model can be mapped directly to real-world data since investment, debt, equity flotations, dividends,
share repurchases, and default are all endogenous. The behavior of the low type in the model is consistent
with empirical observation. In particular, Leary and Roberts (2006) document that “more often than not,
when firms issue equity, they do so before turning to the debt market or in lieu of issuing debt.” Such
behavior is inconsistent with the pecking-order proposed by Myers (1984). In our model, investors reward
the firm for revealing negative information by providing it with ample equity financing and a cash buffer
stock that can be used if investment becomes profitable in the future.
The model is also consistent with stylized facts regarding financial structure dynamics documented by
Fama and French (2002). In simulated data, the leverage ratio is inversely related to lagged profitability
and exhibits mean-reversion. Since the shock ε is public, it can be viewed as a proxy for macroeconomic
shocks.4 Following low realizations of ε, firms have low net worth. In such states, simulated leverage ratios
are particularly high for firms with positive private information regarding the marginal product of capital
since they must send strong signals in conjunction with their major fund-raising efforts. Thus, the model
predicts that firms will have higher average leverage ratios at the tail-end of recessions. This prediction is
4 In contrast, there is no reason to assume insiders are privately informed about the macroeconomy. Hence, θ cannot proxy
for the macroeconomy.
3
consistent with evidence presented by Korajczyk and Levy (2004).
Our theory of the cyclical behavior of leverage is complementary to that proposed by Choe, Masulis
and Nanda (1993). They develop a static model in which firms finance a project of fixed size with debt or
equity, but not both. In their model, equity issuance increases during expansions due to cyclicality in the
investment opportunity set. Our theory is distinct since we hold the investment opportunity set constant
over the business cycle. In our model, cyclical effects are caused by endogenous fluctuations in firms’ internal
funds.
The baseline model also generates predictions consistent with observed patterns of abnormal equity
returns surrounding corporate announcements.
theoretic models with common knowledge.
5
Such announcement effects do not emerge in trade-off
Simulated share prices exhibit positive (negative) abnormal
returns in response to increases (decreases) in capital expenditures. This is consistent with the findings
of McConnell and Muscarella (1985), who document raw equity returns of 1.21% following announced
increases in capital budgets and -1.52% for decreases. The model also predicts that a high debt percentage
in total external finance leads to positive abnormal returns, even when the investment rate is included
as a conditioning variable. Consistent with this prediction, Masulis (1983) documents a 13.97% primary
announcement return in debt for equity exchanges and a -9.91% return in equity for debt exchanges.
In the most closely related paper, Constantinides and Grundy (1990) show that a firm with variable
investment scale can implement first-best if bankruptcy is costless. First-best is achieved with the firm
issuing debt in excess of the amount needed to fund investment. Excess funds are then used for share
repurchases. Our static model, which is of independent interest, alters that of Constantinides and Grundy
(1990) by introducing default costs. Here the firm chooses the least costly mix of real and financial policies
to signal positive private information. This entails issuing risky debt and increasing capital in order to
reduce default costs. In our dynamic model, the hidden information problem repeats and net worth evolves
endogenously. The extension to a dynamic setting adds a number of insights. First, repetition of the hidden
information problem encourages saving and capital accumulation relative to what occurs in the static model.
Second, the continuation value inherent in the firm reduces default costs associated with debt, since the firm
can issue securities against this continuation value in order to meet prior debt obligations. Third, allowing for
endogenous fluctuations in net worth allows the model to explain mean-reverting and countercyclical leverage
ratios, as well as the observed negative relationship between leverage and lagged profitability.6 Finally, by
allowing for endogenous fluctuations in net worth, we can evaluate how the nature of equilibrium (pooling
v. separating) varies endogenously.
Our static model is related to a number of papers that analyze static multi-dimensional signaling.
Milgrom and Roberts (1986) consider multi-dimensional signaling in product markets. Ambarish, John
5 In such environments, only changes in commonly observed state variables (e.g. earnings) induce changes in stock prices.
Given these state variables, investors can predict firm actions, so policy announcements have no effect on prices.
6 A single-period model also generates this prediction if one treats net worth as an exogenous parameter in the comparative
statics. However, net worth is actually an endogenous variable, being determined by lagged policies, e.g. retention decisions.
Endogeneity invalidates comparative statics treating net worth as a parameter.
4
and Williams (1987) and Williams (1988) consider settings in which the firm signals private information
through investment and taxable dividends. There is no debt in these models. In terms of corporate finance
models with multi-dimensional signaling, our static model is most similar to that of Viswanathan (1995),
who also allows signaling through the issuance of debt and investment in a setting where default is costly.
Viswanathan considers a setting where the asymmetric information does not concern the marginal product of
new investment. For example, one may interpret his model as one in which asymmetric information concerns
cash flows from assets in place. In contrast, we consider a setting in which asymmetric information concerns
the marginal product of capital.
Our paper is also related to that of Myers and Majluf (1984), despite the fact that we predict departures
from their asserted pecking-order.
Myers and Majluf present a simple model illustrating that a firm
constrained to finance with equity may rationally forego positive NPV projects if the lemons problem is
sufficiently acute. They briefly (and less formally) consider the choice between debt and equity. They
argue that firms should minimize the informational sensitivity of their securities, which generally implies a
preference for debt over equity.
Nachman and Noe (1994) develop a theoretical foundation for the pecking-order hypothesis. They assume
investment scale is fixed and attention is confined to securities with payoffs that increase monotonically with
cash flow. Under these assumptions, there can be no separating equilibrium, as firms would always benefit
from reporting the highest type, receiving the highest security value (due to monotonicity), and paying
the additional funds as a dividend. Nachman and Noe derive sufficient conditions such that debt will be
the unique source of financing in the pooling equilibrium. The essential idea is that debt minimizes crosssubsidies. The central difference between our setting and that considered by Nachman and Noe is that we
allow the firm to signal using investment scale and share repurchases. The broader, and arguably more
realistic, action space creates the possibility for separating equilibria.
Finally, there has been little work done on models of investment and financing under repeated hidden
information. Lucas and McDonald (1990) develop a dynamic model of investment under hidden information.
However, they constrain the firm to finance with equity. Sannikov (2006) presents a model of optimal security
design when there is one-time hidden information ex ante and repeated moral hazard ex post. Biais, Mariotti,
Plantin and Rochet (2007) and DeMarzo and Fishman (2007) analyze optimal security design when facing
repeated moral hazard. Our focus here is on repeated hidden information.
The remainder of this paper is organized as follows. Section 2 presents our static signaling model. Section
3 characterizes the least-cost separating equilibrium in a dynamic setting. Section 4 uses simulated data to
evaluate empirical implications of the model. Section 5 analyzes the possibility of pooling equilibria. Section
6 extends the model to incorporate a tax advantage to debt. Section 7 concludes.
5
2
Static Signaling Model
2.1
Technology and Timing
In order to convey some of the economic intuition of our dynamic model, we first present the single-period
analog. Comparison of the static and dynamic models sheds light on economic effects and complexities arising
from the problem of repeated hidden information. The static model is also of independent interest in that
it analyzes the optimal mix of costly signals when insiders have private information regarding the marginal
product of capital. This analysis complements the model of Viswanathan (1995) where inside information
concerns cash flows that are additively separable from new investment, e.g. cash flows from assets in place.
Unless stated otherwise, technological assumptions are identical in the static and dynamic models.
There is a risk-free asset paying interest rate r > 0. Agents are risk-neutral and share the discount factor
β ≡ (1 + r)−1 . Capital (k) decays exponentially at rate δ ∈ [0, 1]. The following two assumptions describe
the production technology and timing of information revelation.
Assumption 1. Operating profits are θt εt ktα where α ∈ (0, 1). The private shock θ takes values in
Θ ≡ {θL , θH } with 0 < θL < θH . The public shock ε has a density function f : [ε, ∞) → [0, 1] that is
continuously differentiable with f (ε) > 0 for all ε ≥ ε > 0. In the dynamic model, the private shock follows
a first-order Markov process and the public shock is independently and identically distributed.
Assumption 2. The shock θt is privately observed by an insider-shareholder at the start of the period.
Financing and investment are then determined. At the end of the period, the realized values of ε and net
worth are simultaneously observed by all agents. Net worth is verifiable in a court.
In contrast to a moral hazard model, e.g. DeMarzo and Fishman (2006), the internal resources of the
firm are verifiable by a court. However, we assume the court cannot determine whether a low realization of
operating profits is due to low ε (bad luck) or low θ (negative private information). Thus, it is impossible
to write a θ-contingent financial contract. Despite θ being nonverifiable, Assumption 2 ensures outside
investors can infer the realized value of θt at the end of period t when they observe εt and net worth. In a
dynamic setting, knowing the lagged value of θ is useful to outside investors since it determines transition
probabilities. Effectively, Assumption 2 ensures insiders have a one-period informational advantage relative
to outsiders each period.7
The firm can raise funds by borrowing or issuing new shares. The borrowing technology is a one-period
debt contract with the lender senior in the event of default. The face value of debt for the type-i is denoted
bi .8 Although we do not analyze optimal security design, we here note that if the bankruptcy process is
inherently costly, it would be more efficient to transfer ownership in low profit states using equity warrants
to dilute current shareholders. However, security design should account for tax advantages of debt. If the
7 We
argue that permanent informational advantages are empirically implausible since numerous forces induce revelation of
information about public firms.
8 With some abuse of terminology, we call a firm that has realized the shock θ a type-i firm. However, it must be stressed
i
that the type refers to a temporary shock.
6
debt tax shield is sufficiently large relative to bankruptcy costs, debt will dominate equity warrants as a
device for eliciting truthful revelation. Consistent with this argument, McDonald (2004) shows that written
put options, which have signal value, entail large tax costs relative to synthetic equivalents with explicit
borrowing. Based on this discussion, we incorporate a tax advantage to debt in Section 6.
In the static model, the firm enters the model with an exogenous level of net worth w at the start of the
period. The variable w
e denotes the end-of-period internal resources of the firm provisional on delivering the
promised debt payment. Formally, provisional net worth (w)
e is the sum of capital net of depreciation plus
operating profits less the promised debt payment
w(b,
e k, ε, θ) ≡ (1 − δ)k + θεkα − b.
(1)
In a separating equilibrium, each period’s true θ can be inferred by the investor. To establish separation
it is necessary to consider the payoffs to an insider of type-i who receives the allocation of type-j where j
does not necessarily equal i. Let εdij denote the critical value of ε such that type-i would default given that he
has received the capital stock and taken on the debt obligation of type-j. In the static model, shareholders
default when w
e < 0 and εdij is determined by
w(b,
e k, εdij , θ) = 0 ⇒ εdij =
bj − (1 − δ)kj
.
θi kjα
(2)
If εdij ≤ ε, there is zero probability of default for type-i that has received the allocation of type-j.
In the event of default, the lender seizes the firm’s physical assets and operating profits. In the dynamic
model, the lender also captures the going-concern value of the firm, which is determined endogenously. In
the static model, the lender incurs a legal cost Φ > 0 in the event of bankruptcy. In the dynamic model,
bankruptcy costs are expressed as a percentage of going-concern value. The latter assumption captures the
idea that distress costs are low for tangible assets and high for intangible assets.
As argued by Shyam-Sunder and Myers (1999) and Cooley and Quadrini (2001), there must be some cost
to holding cash or long-lived firms facing financial market imperfections would never pay dividends. Two
real-world costs of hoarding cash are the relatively low after-tax return to saving within the corporate shell
(see Graham (2000)) or agency costs of giving managers access to discretionary cash (see Jensen (1986)).
Presently, we model the latter friction since it is useful to distinguish the proposed theory from trade-off
theory. Taxes are modeled in Section 6. Letting χ be an indicator for b < 0, agency costs of discretionary
cash are equal to χγb2 /2.
Assumption 3 summarizes the firm’s borrowing technology.
Assumption 3. The debt contract is single-period. Default is endogenous and the lender is senior. In the
static model, default costs are Φ > 0. In the dynamic model, default costs are equal to φ times going-concern
value. The yield on corporate saving is r. The firm incurs agency costs equal to χγb2 /2 when it holds cash.
7
Let Ωij denote the expected discounted end-of-period value of shareholders’ equity for a type-i that has
taken the type-j allocation:
Ωij = β
Z
∞
εd
ij
[(1 − δ)kj + θi εkjα − bj ]f (ε)dε.
(3)
Let ρi denote the price of debt issued by type-i in equilibrium:
" Z
ρ ≡ β bi
i
∞
f (ε)dε +
εd
ii
Z
εd
ii
−∞
[(1 − δ)ki +
θi εkiα
#
− Φ]f (ε)dε .
(4)
Shares are issued and repurchased ex dividend. The insider holds m > 0 shares of stock. The number
of shares outstanding at the start of the period, inclusive of insider shares, is c > m. Both c and m are
exogenous. Insider objectives, stated as Assumption 4, are identical to those assumed by Constantinides and
Grundy (1990).9
Assumption 4. The choice of signals is made by a risk-neutral insider-shareholder who maximizes the
cum-dividend value of his equity stake given private information. The insider does not buy additional shares
in the event of an equity flotation and does not tender shares in the event of a share repurchase.
The number of new shares issued is n, with n < 0 indicating a share repurchase. Let s ≡ n/(c + n)
represent the percentage equity stake sold (repurchased). Dividends are denoted d and are constrained to
be nonnegative. Absent this constraint, the firm could avoid signaling costs by having shareholders inject
their own funds directly into the firm. The insider receives a fraction m/c of total dividends and holds an
equity stake of m/(c + n) at the end of the period. Therefore, the cum-dividend value of the insider’s equity
stake is
µ ¶
µ
¶ Z ∞
m
m
[(1 − δ)k + θi εkα − b]f (ε)dε.
d+
β
c
c+n
εd
(5)
It is convenient to think of the insider as proposing an allocation, a ≡ (b, d, k, s), to the investor. Suppose
the insider observes θi at the start of the period. Using the definition of s, the objective function for the
insider (5) simplifies to (m/c)u(b, d, k, s), where
u(b, d, k, s) ≡ d + (1 − s)β
Z
∞
εd
[(1 − δ)k + θi εkα − b]f (ε)dε.
(6)
Conveniently, the multiplicative term m/c has no effect on incentive constraints, since the vector a contains
all relevant choice variables.
9 Myers
and Majluf (1984) also adopt the assumption of controlling shareholder passivity.
8
2.2
The Signaling Game
The insider moves first and offers a feasible allocation to the investor. The set of feasible allocations is A10
A ≡ {(b, d, k, s) : d ≥ 0, k ≥ 0, s ≤ 1}.
The investor then updates his beliefs and either accepts or rejects the allocation. If the investor accepts,
the allocation is implemented. If the offer is rejected, the firm is constrained to finance with internal funds
(i.e. s = 0 and b ≤ 0). The equilibrium concept is perfect Bayesian equilibrium (PBE). A PBE imposes the
following requirements: 1) For any feasible offer, the investor has some belief. 2) The investor’s beliefs are
derived from Bayes’ rule for any offer made in equilibrium. 3) The investor accepts an offer only if it yields
weakly positive profits given his updated beliefs. 4) Each type’s offer is optimal given the lender’s actions.
This formulation of the signaling game assumes investment is contractible, as in the model of Constantinides
and Grundy (1990). In reality, the terms of financing are routinely linked to firms’ stated use of funds. For
example, a secured debt contract links financing to a specific capital investment.
The LCSE depends upon net worth at the start of the period (w). Intuitively, the need for costly signaling
increases in the amount of external funds raised and thus decreases in net worth at the start of the period.
For example, if net worth is sufficiently high, the high type funds investment with internal funds. Therefore,
the low type cannot benefit from mimicry and security mispricing. In such states, there is no need for the
high type to burn money on costly signals.
To derive the LCSE we first solve Program L.
PROGRAM L:
max
a∈A
dL + (1 − sL )ΩLL
subject to the budget constraint
BCL : dL + kL + χγb2L /2 − w ≤ ρL + sL ΩLL .
The left side of BCL measures the amount of funding provided by the investor and the right side represents
the true value of the claims received by the investor. Thus, BCL ensures the investor makes weakly positive
profits in expectation. It should be noted that the budget constraint also accounts for share repurchases.
To see this, assume the type-i insider reveals the firm’s true type by choosing a separating allocation
(bi , di , ki , si ). If the firm issues new shares (n > 0), the equity flotation raises si Ωii . In the event of a
share repurchase, each outsider-shareholder must be indifferent between tendering or not. In this case, si Ωii
also represents the cash outflow from a share repurchase. To demonstrate this, consider a simple example.
Suppose Ωii = 100, c = 50 and n = −10. After the share repurchase, the ex dividend value of each remaining
10 An additional constraint is that the firm cannot repurchase more than c − m shares. That is, s ≥ −(c/m − 1). This
constraint never binds for c/m sufficiently large.
9
share is 100/(50 − 10) = 2.5. Therefore, in order to induce ten shareholders to sell, the firm must pay 25
(= 10 × 2.5). Note si Ωii = −(10/40) × 100 = −25.
We define λi , ηi and ψ i as the multipliers on the budget (BCi ), dividend non-negativity, and equity
∗
constraints (si ≤ 1), respectively. The solution to Program L is denoted a∗L ≡ (b∗L , d∗L , kL
, s∗L ). We next solve
a related Program H.
PROGRAM H:
max
a∈A
dH + (1 − sH )ΩHH
subject to the budget constraint
BCH : dH + kH + χγb2H /2 − w ≤ ρH + sH ΩHH
and a nonmimicry constraint
N MLH : d∗L + (1 − s∗L )ΩLL ≥ dH + (1 − sH )ΩLH .
∗
The solution is denoted a∗H ≡ (b∗H , d∗H , kH
, s∗H ). The multiplier on the nonmimicry constraint is μ.
In solving Program L we do not impose a nonmimicry constraint. Lemma 1 shows this is without loss of
generality.11
Lemma 1. Assume that a∗H and a∗L solve Program H and Program L, respectively, and that s∗L ≥ 0. Then
the high type prefers a∗H to a∗L .
Proof. See Appendix A.
Let b
θ(a) denote the type inferred by the investor conditional upon receiving an arbitrary offer a ∈ A.
As shown in Appendix B, a fully-revealing PBE with allocations (a∗L , a∗H ) can be supported by the following
investor beliefs.
b
θ(a∗H ) = θH ,
b
θ(a∗L ) = θL ,
b
θ(a)
∀a
∈
arg min
b
∈
/
(9)
Z
θ∈Θ
∞
s
Z
∞
εd
θθ
f (ε)dε +
εd
θθ
∗
{aL , a∗H }.
[(1 − δ)k + θεkα − b]f (ε)dε +
Z
εd
θθ
−∞
[(1 − δ)k + θεkα − Φ]f (ε)dε,
On the equilibrium path, the beliefs in (9) are consistent with Bayes’ rule. Off the equilibrium path, the
investor imposes worst-case beliefs in the sense of Brennan and Kraus (1987). In particular, the investor
11 In solving Program L, we may confine attention to s
L ≥ 0 without loss of generality. To see this, note that a share
repurchase by the low type can be replaced with a dividend without affecting the value of the objective function.
10
attaches the lowest possible valuation to any package of securities not issued in equilibrium. For example, if
the firm were to issue shares and/or debt, a worst-case belief imputes type θL . If the firm were to repurchase
shares (and issue no debt), then a worst-case belief imputes type θH .
2.3
Single-Crossing Conditions in Static Model
The marginal effect of b on the expected discounted end-of-period value of shareholders’ equity for a high-type
(low-type) that has taken the high-type allocation is
ΩHH
b
= −β
ΩLH
b
= −β
Z
∞
εd
Z HH
∞
f (ε)dε
(10)
f (ε)dε.
εd
LH
The marginal effect of k on the expected discounted end-of-period value of shareholders’ equity for a high-type
(low-type) that has taken the high-type allocation is
ΩHH
k
= β
ΩLH
k
= β
Z
∞
εd
Z HH
∞
εd
LH
α−1
[1 − δ + αθH εkH
]f (ε)dε
(11)
α−1
[1 − δ + αθL εkH
]f (ε)dε.
In order to obtain indifference curves over alternative allocations, we compute the total differential of the
insider’s objective function u as defined in equation (6). Evaluated at the high type allocation, indifference
curves are defined by
iH
iH
du(aH ; θi ) = ∆d + (1 − sH )ΩiH
× ∆s = 0.
k × ∆k + (1 − sH )Ωb × ∆b − Ω
(12)
From (12), the insider’s willingness to exchange equity for reductions in the face value of debt is
(1 − sH )ΩiH
ds
b
.
(aH ; θi ) ≡
db
ΩiH
(13)
In general, the relative slope of indifference curves in (s, b) space is ambiguous, reflecting competing forces.
On one hand, the low type is more willing to give up equity, since he knows his equity is less valuable. On
the other hand, the high type views servicing debt as more costly since he is less likely to default.
Next, define the left-truncated mean function for the public shock ε:
M (x) ≡ E[ε|ε ≥ x].
The function M plays an important role in the signal content of alternative financial policies. Rearranging
11
equation (13) yields
−(1 − sH )
ds
.
(aH ; θi ) =
α θ M (εd )
db
(1 − δ)kH − bH + kH
i
iH
(14)
Equation (14) tells us that the signal content of a debt-for-equity substitution boils down to which firm has
higher expected equity value, conditional upon survival (denominator). This is ambiguous. On one hand,
the high type has higher equity value for any given draw of ε. On the other hand, the expected value of
ε on the no-default region is higher for the low type. Based on this analysis, we can impose a technical
assumption which is sufficient to ensure satisfaction of a single-crossing condition for financial policy in the
static model.
Assumption 5 (Static Model). The left-truncated mean function for the public shock, E[ε|ε ≥ x], has
elasticity less than one.
As shown below, Assumption 5 is a sufficient condition for indifference curves in (s, b) space to cross
once, with the low type having steeper indifference curves. This is shown as Panel A in Figure 1. Returning
to the discussion above, by limiting the growth rate of the left-truncated mean of ε, Assumption 5 ensures
the high type has higher expected equity value conditional upon survival. This ensures the high type is less
willing to exchange his equity for a debt reduction. Anticipating, other economic effects are present in the
dynamic model, making it more difficult to determine the signal content of debt-for-equity substitutions.
From (12), the insider’s willingness to exchange equity for capital is determined by
ds
(1 − sH )ΩiH
k
.
(aH ; θi ) ≡
dk
ΩiH
(15)
Rearranging terms in equation (15) it is possible to show that in the static model
¯
¯
¯ ds
¯
ds
(1 − sH )α
¯
+ [(1 − α)(1 − δ) + αbH /kH ] × ¯ (aH ; θi )¯¯ .
(aH ; θi ) =
dk
kH
db
(16)
Under Assumption 5, the low type has steeper indifference curves in (s, b) space in the static model. It
follows from equation (16) that the low type is more willing to exchange equity for a unit of capital. That
is, equity-funded capital accumulation is a negative signal in the static model. This is shown as Panel B in
Figure 1. In general, the relative slope of indifference curves in (s, k) space is ambiguous, reflecting competing
factors. On one hand, the low type is more willing to give up equity, since he knows his equity is less valuable.
On the other hand, the high type has a higher marginal product of capital. Under Assumption 5 the first
effect dominates in the static model. Anticipating, other economic effects are present in the dynamic model,
making it more difficult to determine the signal content of equity-funded investment.
We summarize the results of this subsection with the following lemma.
Lemma 2. In the static model, Assumption 5 implies debt-for-equity substitutions are a positive signal and
12
equity-financed investment is a negative signal with
¯
¯
¯
¯
¯ ds
¯
¯
¯
¯ (a; θL )¯ > ¯ ds (a; θH )¯
¯ db
¯
¯ db
¯
ds
ds
(a; θL ) >
(a; θH ) for all a ∈ A such that b ≥ 0.
dk
dk
Proof. From equation (14) it is sufficient to show
θH M (εdHH )
>
−1 −α
−α
θL M (εdLH ) ⇔ θH M [θ−1
H kH (bH − (1 − δ)kH )] > θ L M [θ L kH (bH − (1 − δ)kH )] (17)
−1
⇐ θH M (θ−1
H κ) > θ L M (θ L κ) ∀κ > 0.
Now define a family of functions g(θ) ≡ θM (κ/θ) parameterized by κ > 0. We have
g 0 (θ) = M (κ/θ) − (κ/θ)M 0 (κ/θ).
Assumption 5 ensures g 0 > 0. This establishes the last > sign in (17).¥
2.4
Equilibrium of Static Signaling Game
The full-information first-best investment policy is
kiF B
∙
¸1/(1−α)
αθi E(ε)
≡
.
r+δ
(18)
The following remark provides a description of the full-information economy.
Remark. If the profit shock θi is public information, the firm implements first-best investment (kiF B ) using
a combination of equity and default-free debt. The firm does not hold cash.
The above remark shows that bankruptcy costs are insufficient to induce distortions away from firstbest. If there is symmetric information, the firm can still achieve first-best by financing entirely with equity.
However, if there is hidden information, financing entirely with equity becomes problematic since low types
have an incentive to mimic high types and capture gains from security mispricing.
∗
FB
It is easily verified that the solution to Program L entails kL
= kL
and b∗L = 0. That is, in a static
LCSE there is no need to distort policies of the low type away from the full-information first-best. The
financing of the high type in the LCSE satisfies
¯ ¯
¯¸
∙¯
¯ ¯ ds
¯
∂εdHH
μΩLH ¯¯ ds
d
¯
¯
¯
β
f (εHH )Φ =
¯ db (aH ; θL )¯ − ¯ db (aH ; θH )¯ .
∂bH
λH
(19)
The left side of equation (19) measures marginal bankruptcy costs from debt. The right side measures the
signal content of a debt-for-equity substitution, which was shown to be positive in Lemma 2.
13
The optimality condition for the investment of the high type is
1 =
Z
ε
∞
α
(1 − δ + αθH εkH
)f (ε)dε − β
∂εdHH
μΩLH
f (εdHH )Φ +
∂kH
λH
∙
¸
ds
ds
(aH ; θH ) −
(aH ; θL ) . (20)
dk
dk
The left side of equation (20) measures the marginal cost of investment. The first term on the right side
measures the marginal revenue product of capital. The next term measures the benefit created by capital
investment in terms of reducing bankruptcy costs. The final term on the right side captures the signal
content of equity-funded investment, which was shown to be negative in Lemma 2. Given the competing
effects, it is impossible to determine analytically whether the high type invests more or less than first-best
in the static model.
The following proposition summarizes this discussion and follows directly from the first-order conditions
in Programs L and H.
Proposition 1. In the least-cost separating equilibrium of the static model, the low type chooses first-best
∗
FB
policies, with kL
= kL
and b∗L = 0. If net worth is sufficiently high, the nonmimicry constraint is slack and
∗
FB
the high type also chooses first-best policies, with kH
= kH
and issues safe debt. For lower levels of net
worth, the high type issues defaultable debt and distorts investment away from first-best.
3
Dynamic Signaling Model
There are numerous reasons to consider a dynamic model. The foremost reason for doing so is that it
gives rise to new economic effects. As shown below, repeated hidden information results in an endogenous
precautionary motive for reducing debt, retaining cash and investing in more physical capital. This same
precautionary motive also alters the signal content of alternative policies. A second motive is technical,
since we here develop a tractable algorithm for incorporating hidden information into a dynamic neoclassical
investment framework. A third motive for considering a dynamic setting is increased realism. For example, it
is shown below that the static model overstates the probability of default by ignoring firms’ ability to roll-over
debt. Fourth, we can calibrate the dynamic model to real-world panel data in order to infer the magnitude
of the costs arising from asymmetric information and to assess the model’s ability to match moments. Fifth,
the dynamic calibrated model represents a robust laboratory for generating null-hypotheses (see Hennessy
and Whited (2005)). Finally, the dynamic model allows us to analyze the effect of shock persistence.12
12 A two-period model would achieve some of these objectives. However, such a model is equally complex as an infinite-horizon
model since both involve continuation value functions with no closed-form solutions.
14
3.1
Technology
For the dynamic model, we retain Assumptions 1-4, but drop Assumption 5. Anticipating, we impose a
stronger assumption than Assumption 5 in the dynamic model in order to obtain a single-crossing condition.
We let πij denote the probability of θi conditional on lagged type θj . The shock θ is persistent with
1 > π HH ≥ π HL > 0.
At the end of each period, after the public shock is observed, the firm either defaults or delivers the
promised debt payment. Default occurs if and only if current shareholders will be unable to raise a sufficient
amount of internal and external funds in the next round of financing in order to deliver the promised amount
bi . Consistent with the absolute priority rule, in the event of default the lender demands a cash payment
that pins existing shareholders to their reservation value of zero. Anticipating, this is accomplished by
forcing existing shareholders to sell off the firm and having them deliver to the lender the cash received in
consideration for the equity. Cooley and Quadrini (2001) consider a similar debt contract, but they allow
shareholders to inject cash directly into the firm. Anticipating, hidden information substantially alters the
analysis of endogenous default.
The variable w
e again denotes provisional net worth, i.e. the net worth of the firm if shareholders deliver
the promised debt payment. Following Cooley and Quadrini (2001), we assume capital investment is perfectly
reversible. Thus, we need to keep track of total internal resources, but do not need to maintain separate state
variables for cash and capital. There are two state variables in the model: the lagged value of θ and net worth
(w). In general, there are two equity value functions, with vi (w) defined as the market value of shareholders’
equity given that the firm began the period with type θi and ended the period with net worth w. The
type at the beginning of the period is relevant to equity value since the lagged type determines transition
probabilities. If the shock θ is i.i.d. there is no need to keep track of the lagged type as a state variable and
vH = vL ≡ v. It is worth stressing that the equity value functions reflect the value of shareholders’ equity
after observing net worth but prior to observing the next realized value of θ.
There exists an endogenous cutoff wd < 0 such that shareholders must default if w
e < wd . If w
e ≥ 0, default
does not occur since the promised debt payment can be delivered using internal resources. If w
e ∈ [wd , 0),
shareholders deliver the promised debt payment using a portion of the external funds raised in the next
financing round. In fact, if w
e = wd , shareholders can only deliver the promised debt payment by selling off
all the equity in the firm (s = 1). As verified below, the model has an endogenous default condition of the
form
vL (wd ) = vH (wd ) = 0.
(21)
If w
e < wd , default occurs and the lender demands the maximum cash payment consistent with limited
liability. From equation (21) we know the lender will demand a cash payment that leaves current shareholders
15
with net worth equal to wd in the event of default. The resulting law of motion for net worth is
w(b, k, ε, θ) = max{wd , w(b,
e k, ε, θ)}.
(22)
¯ ¯
The domain for the two equity value functions is W ≡ [wd , ∞). We heuristically speak of ¯wd ¯ as representing
¯ ¯
the firm’s going-concern value. More precisely, it is shown below that ¯wd ¯ is equal the amount for which
the firm can be sold to an uninformed outside investor.
In the dynamic model shareholders default when w
e < wd , implying that the default-inducing shock εdij
is determined by
w(b,
e k, εdij , θ) = wd ⇒ εdij ≡
bj − (1 − δ)kj + wd
.
θi kjα
(23)
If εdij ≤ ε, there is zero probability of default for type-i that has received the allocation of type-j. As discussed
above, an important difference between the static and dynamic models is that the option value inherent in
the firm makes default less likely in the latter, ceteris paribus. This can be seen by comparing the default
thresholds in equations (2) and (23).
In the event of default the lender demands a revised debt payment from shareholders, call it bri < bi , that
leaves them with net worth equal to wd . Therefore, we compute bri using
¯ ¯
(1 − δ)ki + θi εkiα − bri = wd ⇒ bri = (1 − δ)ki + θi εkiα + ¯wd ¯ .
(24)
Equation (24) indicates that the lender seizes the firm’s physical assets, operating profits, and going-concern
value. Since a fraction (φ > 0) of going-concern value is spent on legal fees, the lender’s net default recovery
is
¯ ¯
bri − φ|wd | = (1 − δ)ki + θi εkiα + (1 − φ) ¯wd ¯ .
(25)
Let Ωij denote the expected discounted end-of-period value of shareholders’ equity for a type-i that has
taken the type-j allocation. In a dynamic setting
Ωij ≡ β
Z
∞
εd
ij
vi [(1 − δ)kj + θi εkjα − bj ]f (ε)dε.
(26)
A comparison of equations (26) and (3) reveals the fundamental difference between the dynamic and static
models. In the dynamic model, the value of equity is determined by some endogenous transformation (vi )
of internal resources. In a static model, equity value is linear in internal resources.
The market price of the debt issued by type-i in the dynamic equilibrium is
" Z
ρ ≡ β bi
i
∞
εd
ii
f (ε)dε +
Z
εd
ii
−∞
[(1 − δ)ki +
θi εkiα
d
#
+ (1 − φ)|w |]f (ε)dε .
(27)
The informed insider maximizes the cum-dividend value of his equity stake. In the dynamic model, the
16
objective function of the informed insider of type-i is
µ ¶
µ
¶ Z ∞
m
m
vi [(1 − δ)k + θi εkα − b]f (ε)dε.
d+
β
c
c+n
εd
(28)
Using the definition of s, the objective function for the insider (28) simplifies to (m/c)u(b, d, k, s) where
we have redefined u as:
u(b, d, k, s) ≡ d + (1 − s)β
Z
∞
εd
vi [(1 − δ)k + θi εkα − b]f (ε)dε.
(29)
Comparison of equations (29) and (6) reveals the key difference between the static and dynamic models in
terms of an insider’s objective function. In the dynamic model, the equity value and objective function is
determined by an endogenous transformation (vi ) of internal resources. The insider is forward-looking by
construction. In particular, the equity value functions (vL , vH ) entering the insider’s objective function are
constructed to correctly capitalize the outcome of all future signaling games.
3.2
Repeated Signaling Model
In the repeated signaling game, the insider again moves first and offers an allocation to the investor. The
investor then updates his beliefs and either accepts or rejects the offer. If the investor accepts, the allocation
is implemented. If the offer is rejected, the firm is then constrained to finance with internal funds (i.e.
s = 0 and b ≤ 0). To derive the LCSE we again solve Programs L and H as defined in Section 2. However,
the dynamic model is more complex since the budget and nonmimicry constraints involve unknown equity
value functions. Conveniently, the equity value functions can be defined recursively. In order to derive the
end-of-period value of total shareholders’ equity, we begin by noting that “next period’s insider” holds a
fraction equal to m/c of all shares. Therefore, the total value of all shares outstanding is simply c/m times
the expectation (over types) of u given in equation (29). Recalling that the realized type is a Markov process,
we have
vj (w) ≡ π Hj
"
d∗H
+ (1 −
+(1 − π Hj )
"
d∗L
s∗H )β
Z
+ (1 −
∞
εd
HH
vH [(1 −
s∗L )β
Z
∞
εd
LL
∗
δ)kH
vL [(1 −
+
∗ α
θH ε(kH
)
∗
δ)kL
+
−
b∗H ]f (ε)dε
∗ α
θL ε(kL
)
−
#
(30)
#
b∗L ]f (ε)dε
.
Although w is omitted as an argument in the allocation vectors, it is worth stressing that the equilibrium
allocations (a∗L , a∗H )(w) depend upon net worth at the start of the period.
The final step in the construction of the dynamic LCSE is the determination of the minimal level of net
worth, denoted wd . We first note that BCL is binding in the LCSE. Substituting the budget constraint into
17
the maximand, we can rewrite Program L as
Program L : max
bL ,kL
w + ρL (bL , kL ) + ΩLL (bL , kL ) − kL − χγb2L /2.
Since the objective function for Program L is linear in net worth, there exists a w
b at which the maximized
objective in Program L is just equal to zero. For any w < w,
b the limited liability of shareholders would be
violated. Next note that the maximized objective in Program H is necessarily equal to zero at this same level
of net worth w.
b To see this, note that any allocation with dH > 0 or sH < 1 would violate the nonmimicry
constraint. It follows that regardless of the realized type, shareholders achieve a continuation value of zero
if they start a financing round with net worth equal to w.
b Therefore, in the baseline model, where attention
is confined to LCSE, the default threshold is wd = w,
b where
w
b ≡ − max
bL ,kL
ρL (bL , kL ) + ΩLL (bL , kL ) − kL − χγb2L /2.
(31)
As discussed above, equation (31) shows that the default threshold for net worth is equal to the value for
which ownership of the firm can be sold to an uninformed investor. Finally, we note that deadweight signaling
costs shift down the equity value functions and limit the amount of external equity the firm can raise in
order to meet prior debt obligations. Thus, our model predicts early defaults relative to structural models
of endogenous default that adopt the assumption of symmetric information.
3.3
Dynamic Least-Cost Separating Equilibrium
In order to express the optimality conditions compactly, we redefine some variables. The marginal effect of
b on the expected discounted end-of-period value of shareholders’ equity for a high-type (low-type) that has
taken the high-type allocation is
ΩHH
b
≡ −β
ΩLH
b
≡ −β
Z
∞
εd
Z HH
∞
εd
LH
0
α
vH
[(1 − δ)kH + θH εkH
− bH ]f (ε)dε
(32)
0
α
vL
[(1 − δ)kH + θL εkH
− bH ]f (ε)dε.
The marginal effect of k on the expected discounted end-of-period value of shareholders’ equity for a high-type
(low-type) that has taken the high-type allocation is
ΩHH
k
ΩLH
k
≡ β
≡ β
Z
∞
εd
HH
Z
∞
εd
LH
α−1
0
α
vH
[(1 − δ)kH + θH εkH
− bH ][1 − δ + αθH εkH
]f (ε)dε
(33)
α−1
0
α
vL
[(1 − δ)kH + θL εkH
− bH ][1 − δ + αθL εkH
]f (ε)dε.
Let μ(·) denote the wealth-contingent multiplier on the N MLH constraint in Program H.
In order to interpret the optimality conditions, it is useful to determine the shadow value of internal
18
resources. This is the subject of Lemma 3.
Lemma 3. In the dynamic model, there exists a level of net worth, wslack , such that
vi0 (w) = 1 + πHi × μ(w) × [ΩHH (w) − ΩLH (w)]/ΩHH (w) > 1 for w ∈ [wd , wslack )
(34)
= 1 for w ≥ wslack .
Proof. See Appendix A.
The intuition for Lemma 3 is as follows. If the realized type in the subsequent period is θL , a dollar of
internal funds is just worth a dollar, as the firm essentially obtains financing on fair terms. However, if the
realized type in the subsequent period is θH , a dollar of internal funds can be worth more than a dollar if it
allows the firm to avoid signaling costs. This explains why the shadow value of internal resources depends
upon the probability (π Hi ) of transitioning to the high type. Note also that the adverse selection problem is
manifest in equation (34) with the term (ΩHH − ΩLH )/ΩHH measuring relative equity valuations. Lemma 3
indicates that repeated hidden information results in pseudo-risk-aversion, since the shadow value of internal
funds is above one for low net worth levels and is equal to one if net worth is sufficiently high.13 Intuitively,
if net worth is low and the probability of transitioning to a high type is high, then the marginal value of
internal funds is very high since internal funds reduce expected signaling costs in the next financing round.
The next proposition follows from the first-order conditions for the debt and capital of the low type. The
calculations are included in Appendix C.
Proposition 2. In the least-cost separating equilibrium of the dynamic model, the low type chooses second∗
SB
FB
best policies. For all w ∈ [wd , ∞), kL
(w) = kL
> kL
and b∗L (w) = bSB
L < 0 where
bSB
L =−
β
γ
∙Z
ε
∞
¸
0
SB
SB α
[vL
((1 − δ)kL
+ θL ε(kL
) − bSB
)
−
1]f
(ε)dε
.
L
(35)
Dividends and equity issuance for the low type are contingent upon net worth with
w
∗
< kL
− βb∗L + γ(b∗L )2 /2 ⇒ d∗L = 0 and s∗L > 0
w
∗
∗
≥ kL
− βb∗L + γ(b∗L )2 /2 ⇒ d∗L = w − (kL
− βb∗L + γ(b∗L )2 /2) and s∗L = 0.
The intuition for the low type policies is as follows. In order to discourage imitation by the low type,
the LCSE makes the low type as well off as possible. In the static signaling model, the optimal policy
gives the low type the full-information first-best allocation. In the dynamic model, the low type is given a
second-best allocation which accounts for the fact that the shadow value of internal resources exceeds one.
Consequently, the low type overinvests relative to first-best and maintains costly financial slack. It is also
13 We
verify global concavity of the value functions using numerical simulations.
19
∗
worth emphasizing that both b∗L and kL
are invariant to net worth. Therefore, the low type satisfies BCL
by varying dividends and equity issuance only. Note, this is the exact opposite of the static pecking-order
hypothesis that debt (and only debt) is used to achieve budget balance.
In Appendix C it is shown that the high type will not pay a dividend if the nonmimicry constraint
(N MLH ) binds. Intuitively, when the nonmimicry constraint binds, the high type must engage in costly
signaling in order to raise external funds. In order to minimize the external funding requirement, zero
dividends are paid when the nonmimicry constraint binds. In the LCSE, b∗H is determined by
#
∂εdHH
d
d
− δ)kH +
− bH ) − 1]f (ε)dε +
f (εHH )φ|w | + χγbH
β
∂bH
εd
HH
µ LH ¶
∙ HH
¸
μΩ
Ωb
ΩLH
b
=
− LH .
[1 − sH ]
λH
ΩHH
Ω
"Z
∞
0
[vH
((1
α
θH εkH
(36)
0
The term vH
− 1 on the left side of (36) captures a novel deadweight cost of debt service in a dynamic
setting with repeated hidden information. The bondholder values a dollar of debt service at one dollar.
However, anticipation of future signaling costs causes shareholders to view the cost of debt service as weakly
greater than one. The next term on the left side of the equation measures marginal default costs. The third
term measures the marginal cost of saving. The right side measures the signal content of debt-for-equity
substitutions, discussed in greater detail below. From (36) and Lemma 3 it follows that μ = 0 ⇒ bH < 0.
Condition (36) therefore leads to a key implication of the model, that signaling must be a concern if the firm
issues debt.14
∗
The optimality condition pinning down kH
is
1=β
+β
+
"Z
"Z
µ
∞
εd
HH
0
[vH
((1
εd
HH
−∞
μΩLH
λH
[1 − δ +
¶
− δ)kH +
α
θH εkH
− bH )][1 − δ +
α−1
αθH εkH
]f (ε)dε
(1 − sH )
∙
∂εd
+ HH f (εdHH )φwd
∂kH
¸
ΩHH
ΩLH
k
k
−
.
ΩHH
ΩLH
#
α−1
αθH εkH
]f (ε)dε
#
(37)
Equation (37) states that the optimal investment policy equates marginal benefits with the unit price of
capital. The first term on the right side of the equation measures the benefit to shareholders from an
additional unit of installed capital. Notice that installed capital has an added precautionary benefit when
0
vH
> 1. Intuitively, capital investment has added value in that it increases the pool of future internal
resources. The second term measures the benefit of investment accruing to bondholders. The last term
measures the signal content of equity-financed investment, which is discussed in greater detail below.
Proposition 3 spells out some implications of the first-order conditions when the nonmimicry constraint
is slack.
14 This
result no longer holds once we introduce tax advantages to debt finance.
20
Proposition 3. In the least-cost separating equilibrium of the dynamic model, for all w ≥ wslack , the high
∗
SB
FB
type implements second-best policies with kH
(w) = kH
> kH
and b∗H (w) = bSB
H < 0 where
bSB
H
β
=−
γ
∙Z
∞
0
[vH
((1
ε
−
SB
δ)kH
+
SB α
θH ε(kH
)
−
bSB
H )
¸
− 1]f (ε)dε .
(38)
For w ≥ wslack , dividends and equity issuance are contingent upon net worth with
w
SB
SB 2
∗
∗
∈ [wslack , kH
− βbSB
H + γ(bH ) /2) ⇒ dH = 0 and sH > 0
w
SB
SB 2
∗
SB
SB
SB 2
∗
∈ [kH
− βbSB
H + γ(bH ) /2, ∞) ⇒ dH = w − [kH − βbH + γ(bH ) /2] and sH = 0.
Consider next w < wslack , where the nonmimicry constraint is binding. In terms of indifference curves,
the debt optimality condition for the high type (36) can be rewritten as
"Z
#
∂εdHH
d
d
β
− δ)kH +
− bH ) − 1]f (ε)dε +
f (εHH )φ|w | + χγbH
∂bH
εd
HH
¯ ¯
¯¸
µ LH ¶ ∙¯
¯ ¯
¯
¯ ds
μΩ
¯ (aH ; θL )¯ − ¯ ds (aH ; θH )¯ .
=
¯
¯
¯
¯
λH
db
db
∞
0
[vH
((1
α
θH εkH
(39)
Equation (39) tells us that the high-type’s borrowing depends upon the signal content of debt-for-equity
substitutions, as measured by the difference in the slope of the indifference curves in (s, b) space. If debt has
positive signal content, the high type borrows more/saves less in order to separate from the low type.
Intuition suggests that four factors determine the relative slopes of the types’ indifference curves in (s, b)
space. First, the low type knows his equity is less valuable, which increases his willingness to exchange equity
for debt reductions. Second, the low type may have a stronger precautionary motive for avoiding debt since
he will have lower net worth for all possible realizations of the public shock. However, persistence of the
private shocks may induce the higher type to have a stronger precautionary motive. Finally, the high type
will service debt in more states of nature, increasing his marginal cost of debt service.
A bit of calculation allows us to write
h
i
R∞ 0
f (ε)
)
1
+
(v
−
1)
dε
−(1
−
s
d
H
d
i
εiH
ds
1−F (εiH )
.
(aH ; θi ) =
R∞
α
d
db
(1 − δ)kH − bH + kH θi M (εiH ) + εd (vi − w) 1−Ff (ε)
dε
d
(ε )
iH
(40)
iH
Comparison of equations (40) and (14) reveals that the signal content of debt-for-equity substitutions is more
complicated in the dynamic model since the slope of the indifference curves depend upon the properties of
endogenous value functions. We see that the numerator of (40) involves an added term (vi0 − 1) capturing
the marginal precautionary value of internal resources. The denominator of (40) also involves an added term
(vi − w) capturing the enterprise value of the firm. The enterprise value of a firm is traditionally defined as
21
the difference between firm value and its cash balance. Since physical capital is perfectly reversible in the
model, a dollar in cash and a dollar in capital have equivalent implications for equity’s continuation value.
Thus, enterprise value in our model is equal to vi − w. The presence of additional terms in (40) is intuitive.
Since the numerator of (40) measures the marginal cost of debt service, it must account for the precautionary
value of internal funds. Since the denominator measures equity value, it must account for enterprise value.
For the remainder of the paper, we adopt Assumption 50 .
Assumption 50 (Dynamic Model). The public shock is exponentially distributed.
Under the exponential distribution f (ε) ≡ ξe−εξ . Assumption 50 is stronger than Assumption 5, since the
elasticity of the left-truncated mean function at an arbitrary cutoff point x is equal to x/(x+ξ −1 ) < 1. Under
the exponential distribution, we obtain unambiguous results regarding the signal content of debt-for-equity
substitutions allowing for arbitrary correlation in the private shocks.
We begin by rewriting the expression for the marginal cost of debt (32) as follows
Ωij
b =
−β
θi kjα
Z
∞
εd
ij
f (ε)
µ
¶
∂
vi ((1 − δ)kj + θi εkjα − bj ) dε.
∂ε
(41)
Using integration by parts it follows that
Ωij
b =
−ξΩij
.
θi kjα
(42)
It follows that a debt-for-equity substitution is a positive signal, with
¯
¯ ¯
¯
∙ HH
¸
−1
LH
¯ ds
¯ ¯
¯
(1 − sH )ξ(θ−1
L − θH )
¯ (a; θL )¯ − ¯ ds (aH ; θH )¯ = (1 − sH ) Ωb − Ωb
.
=
α
¯ db
¯ ¯ db
¯
HH
LH
Ω
Ω
kH
(43)
We have thus established Proposition 4. The Corollary follows directly from the optimality condition (39)
and tells us that the high type will issue more debt when w ∈ [wd , wslack ) than when w ≥ wslack .
Proposition 4. If the public shocks have the exponential distribution, debt-for-equity substitutions are a
positive signal in the dynamic model with
¯
¯ ¯
¯
¯ ds
¯ ¯
¯
¯ (aH ; θL )¯ > ¯ ds (aH ; θH )¯ .
¯ db
¯ ¯ db
¯
Corollary. For w ∈ [wd , wslack ), the high type chooses b∗H (w) > bSB
H .
We can also rewrite the optimality condition for the high-type capital stock (37) using indifference curve
notation as
1=β
+β
"Z
"Z
∞
εd
HH
εd
HH
−∞
0
[vH
((1
− δ)kH +
α
θH εkH
− bH )][1 − δ +
#
α−1
αθH εkH
]f (ε)dε
(44)
# µ
¸
¶∙
d
μΩLH
∂ε
ds
ds
α−1
d
d
HH
[1 − δ + αθH εkH ]f (ε)dε +
f (εHH )φw +
(aH ; θH ) −
(aH ; θL ) .
∂kH
λH
dk
dk
22
The optimality condition above tells us that the high-type’s capital stock increases with the signal content of
equity-financed investment, as measured by the difference in the slope of the two type’s indifference curves
in (s, k) space. Intuition suggests a number of factors determine the relative slopes of indifference curves in
(s, k) space. First, the high type generates more future cash than a low type for a given level of capital,
which increases his willingness to exchange equity for capital. Second, the low type has less valuable equity,
increasing his willingness to trade equity for capital. In a static setting, the second effect dominates when
ε has the exponential distribution. However, precautionary motives complicate the analysis. Persistence
in θ implies that a high type has a high probability of being a high type in the subsequent period. For
such a firm, the future internal cash generated by installed capital may be more valuable since internal funds
reduce exposure to signaling costs. This effect serve to increase the slope of the high-type indifference curves.
However, the low type may place a higher shadow value on a marginal dollar at the end of the period, since
it necessarily realizes lower net worth.
An efficient mix of real and financial signals equates the ratio of distortion to signal content at the margin.
In particular, equations (44) and (39) imply that in the LCSE
1−β
β
=
hR
∞
hR
∞
εd
HH
εd
HH
α−1
0
(vH
) ∗ [1 − δ + αθH εkH
]f (dε) +
R εdHH
−∞
ds
dk (aH ; θ H )
∂εd
i
−
α−1
[1 − δ + αθH εkH
]f (dε) +
ds
dk (aH ; θ L )
0
[(vH
) − 1]f (dε) + ∂bHH
f (εdHH )φ|wd | + χγbH
H
¯ ds
¯ ¯
¯
.
¯ (aH ; θL )¯ − ¯ ds (aH ; θH )¯
db
db
∂εd
d
d
HH
∂k f (εHH )φw
i
(45)
The numerator of the previous equation measures the deviation of capital from second-best (where secondbest accounts for precautionary value and default costs). The numerator of the right side measures the
deviation of debt from second-best. The denominators measure the marginal information content of real and
financial signals, respectively.
4
Simulation of Baseline Model
The following parameter values are used in the simulations: β = 1/1.065; α = 0.60; δ = 0.10; γ = 0.05;
φ = 0.20; θH = 0.8; θL = 0.6; and π HL = π HH = 1/2. The public shocks are exponentially distributed with
f (ε) = 2 exp(−2ε). The procedure used to solve the model numerically is described in Appendix D. Once
the model is solved, the wealth-contingent policy functions (a∗L , a∗H ) : W → A × A are used to generate a
simulated panel data set. We draw 3000 sample paths of public and private shocks consisting of 31 draws
for each. All firms start with zero net worth and the initial period is dropped from the sample. We then use
the policy functions generated by the model to determine shock-contingent policy and wealth paths. The
simulated panel data set is similar in size to those commonly used in empirical testing.
To illustrate concavity of the equity value function, or pseudo-risk-aversion, we begin by plotting the firm’s
enterprise value. In the present setting, with independent draws of θ, the enterprise value (e) is computed
23
as e(w) = v(w) − w. The slope of the enterprise value function is a direct measure of the precautionary
motive for saving, since e0 = v0 − 1. The slope of the enterprise value function measures the marginal net
gain current shareholders would capture if they could inject cash directly into the firm. We can express the
enterprise value as
¯ ¯
e(w) = ¯wd ¯ +
Z
w
wd
[v 0 (w) − 1]dw
∀ w > wd .
(46)
Thus, the enterprise value, at any point in the wealth space is equal to the going-concern value plus the
cumulative precautionary value of internal resources. Figure 2 plots the enterprise value function. Consistent
with concavity of the equity value function (v), the slope of the enterprise value function is positive but
decreasing in net worth. The enterprise value has a slope of zero for high levels of net worth where the
nonmimicry constraint is slack. Examining figure 2, we see that starting at the default threshold, the
value gained from a one unit cash infusion is roughly equal to 1.05. This magnitude is sensitive to the
parameterization of the model. For example, the gain from cash infusions increases in the probability of the
high type.
Figure 3 plots the capital allocations of each type relative to first-best. In the LCSE, the low type
invests slightly above first-best regardless of realized net worth. This overinvestment reflects the fact that
informational asymmetries create a precautionary motive for capital accumulation. The high type also invests
more than first-best, with the extent of the distortion decreasing in net worth. When net worth is low, the
nonmimicry constraint binds and the high type overinvests by a larger amount in order to reduce default
costs. If net worth is sufficiently high, the nonmimicry constraint is slack, but the high type still overinvests
due to precautionary motives.
Figures 4 plots the wealth-contingent financing policies for each type. Consistent with Proposition 2, the
low type uses dividends and equity issuance as the sole means of achieving budget-balance, while retaining
a wealth-invariant level of savings. When net worth is low, the low type sets the dividend to zero and issues
a large amount of equity. Equity issuance for the low type then declines monotonically in net worth.
The debt of the high type declines monotonically in net worth. Intuitively, increases in net worth reduce
the need for external financing. This allows the high type to cut back on costly signals. Therefore, after
a high realization of the public (ε) shock, the model predicts that firms will choose low leverage ratios.
Thus, the model is consistent with the empirical observation that leverage ratios are countercyclical (e.g.
Korajczyk and Levy (2004)). This is also consistent with the observation that leverage ratios are decreasing
in lagged profitability (e.g. Fama and French (2002)).
It is worth noting that when the high type issues equity, it almost always conducts a joint offering which
combines equity with debt. In contrast, the low type issues equity without any debt. This is consistent
with existing studies. Asquith and Mullins (1986) document negative abnormal returns in a sample of
pure common stock offerings. Masulis and Korwar (1986) find that seasoned equity offerings are associated
with negative price changes on average. However, the announcement return is positively related to leverage
changes.
24
Turning next to dividend policy, we see that both types pay dividends only if net worth is sufficiently
high. This prediction is consistent with the empirically observed positive relation between dividends and
internal funds. It is also apparent that the low type initiates dividends at a lower net worth threshold than
the high type. This difference reflects the fact that the low type has inferior investment opportunities.
Table 1 reports reduced-form leverage regressions similar to those reported in the empirical literature.
In the first three regressions, the dependent variable is the book leverage ratio. In the last regression, the
dependent variable is the change in the book leverage ratio. The first row utilizes a specification from ShyamSunder and Myers (1999). Shyam-Sunder and Myers use this regression to test the pecking-order hypothesis
that debt is used to fill all financing gaps. In our model, the financing gap is k + d − w. According to the
pecking-order as traditionally specified, the predicted coefficient on the (capital-normalized) financing gap is
one. Inspecting Table 1, we see that the simulated firms fill only 24% of the financing gap with debt. Leary
and Roberts (2006) show that the test constructed by Shyam-Sunder and Myers is prone to Type II errors.
Based on this simulated regression, we argue that the statistical test proposed by Shyam-Sunder and Myers
is also prone to a form of Type I error. In our simulated data, the null hypothesis that “hidden information
influences financing decisions” would be incorrectly rejected if the econometrician were to assume that the
coefficient on the financing gap should be one.
The next two regressions are similar to those reported by Fama and French (2002). Fama and French
report a negative relationship between leverage and lagged profits. In the simulated data, we also find that
leverage is declining in lagged measures of profitability. In the model, high realizations of the public shock
ε result in high lagged profitability and high net worth. Firms with positive information use the increase
in net worth to reduce their reliance on debt. Also consistent with the estimates of Fama and French, the
coefficient on the market-to-book ratio is positive in the simulated data.
Shyam-Sunder and Myers (1999) and Fama and French (2002) both estimate variants of the following
equation to test for mean-reversion in leverage
LEVt − LEVt−1 = B0 + BT A × [LEV ∗ − LEVt−1 ] + ut .
(47)
We follow Shyam-Sunder and Myers in using the firm-specific sample average leverage ratio as an estimate
of the “target” leverage ratio. Shyam-Sunder and Myers report a significant estimate of BT A = 0.41. Fama
and French (2002) estimate smaller, yet significant, coefficients with BT A in the range of 0.07 to 0.10. These
regressions are of interest since the empirical literature often treats mean-reverting leverage as prima facie
evidence in favor of trade-off theory with transactions costs. For example, Fama and French (2002) state
that, “the simple pecking order predicts that... the speed of adjustment is indistinguishable from zero,
whereas the trade-off model says it is reliably positive.” In the fourth regression presented in Table 2, the
estimated value of BT A is 0.18. Thus, the simulated firm exhibits mean-reversion speeds consistent with
those actually observed—absent taxes or direct transactions costs.
25
The model provides a laboratory for analyzing the cross-sectional determinants of abnormal returns
associated with investment and financing announcements. We begin by noting that this analysis is not
always directly comparable to existing empirical studies. In the simulated data, we can measure the stock
price after the market has fully incorporated information about lagged earnings. This ex ante stock price
is equal to v(w)/c. We can then measure the stock price just after the firm announces its financing plans.
This ex post stock price is equal to [d + (1 − s)Ω]/c. The pure abnormal return (AR) associated with the
announced financing is the ex post price less the ex ante price normalized by the ex ante price
ARt ≡
dit + (1 − sit )Ωii
t − v(wt )
.
v(wt )
(48)
In contrast, empirically observed abnormal returns may reflect market inferences about lagged earnings in
addition to inferences about future prospects.
Table 2 analyzes the cross-sectional determinants of abnormal returns by treating the simulated ARt
as a dependent variable. Such regressions are common to the event study literature. A common theme
running through the regressions is that high investment rates are statistically and economically significant
predictors of abnormal returns. This is consistent with the empirical evidence presented by McConnell and
Muscarella (1985) that increases in capital expenditures are associated with positive abnormal returns. In
the first reported regression, the abnormal return is positively related to the investment rate. In the second
and third regressions we see a high percentage of debt in total external financing is also associated with
positive abnormal returns. This is consistent with Masulis’ (1983) finding that debt-for-equity exchanges
are associated with positive abnormal returns. In the fourth regression we see that, unconditionally, the
abnormal return is increasing in leverage. However, the fifth regression reveals that the significance of
leverage is sensitive to the regression specification. In particular, the leverage ratio becomes insignificant
once we include the investment rate as a regressor. This is because the high type always invests much more
than the low type in our model. However, the high type does not necessarily issue much more debt. For
example, when net worth is sufficiently high both firms actually save. Therefore, capital expenditures are
the best predictor of abnormal returns.
5
Pooling Equilibrium
The objective of this section is to determine whether it is possible for the two firm types to pool at the
same allocation, say aP (w), at some points in the state (w) space. To this end, we rely on results derived
by Maskin and Tirole (1992) and Tirole (2006), who consider an option contract game.15 The utility of the
option contract game is that it allows one to narrow the set of possible equilibria.
The timing and equilibrium concept in the option contract game are the same as that for the dynamic
signaling game. However, the initial offer is a menu rather than an allocation. In particular, the game begins
15 The
exposition closely follows Tirole’s Section 6.4.
26
with the insider proposing to the investor an option contract consisting of a pair (a1 , a2 ) ∈ A × A. Formally,
the contract represents an agreement by the two parties to enter into a direct revelation mechanism. If he
accepts, the investor has entered into a binding agreement to fund the firm regardless of which option on the
menu the insider subsequently chooses. In reality, one can view a shelf-registration as resembling an option
contract since a shelf-registration narrows the menu of securities from which a firm may (costlessly) select
after it acquires new information.
We look for PBE at each point in the net worth space. Importantly, each point in the net worth space
has measure zero. Therefore, at each point in the state space the agents can treat the value functions as
given. Similarly, we the modelers can treat the value functions as fixed even as we vary the equilibrium
at particular points. Having said this, all discussions regarding equilibrium at particular points in the net
worth space should be understood as being conditioned on internally consistent value functions. Recursive
methods are used to ensure internal consistency.
We begin by noting that for all w ≥ w,
b the LCSE entailed s∗L (w) ≥ 0. Therefore, for all w ≥ w,
b the
investor would earn nonnegative profits if the high type took the low type’s allocation. Thus, Tirole’s weak
monotonic profit condition is satisfied and the pair (a∗L , a∗H )(w) represents the Rothschild-Stiglitz-Wilson
allocation (RSW). For w < w,
b the constraint BCL cannot be satisfied by any a ∈ A. Thus, for w < w
b < 0,
both types get RSW payoffs of zero. We refer to these RSW payoffs as the “separating payoffs.”
The following is a restatement of Tirole’s Proposition 6.1.
Lemma 4. The separating payoffs are always in the equilibrium set. The payoffs from pooling at an option
contract (aP , aP ) are in the equilibrium set only if the investor earns a weakly positive profit in expectation
(based upon his prior beliefs) and the payoffs Pareto dominate the separating payoffs from the perspective of
both insider types.
Recall that when attention was confined to separating equilibria, the default threshold was given by
d
w =w
b as defined in equation (31). We conjecture, and verify, that the default threshold in the present
setting, which allows for pooling, is wd = w
bP < w.
b That is, the possibility of pooling increases the
continuation region relative to what is possible if only separating equilibria are considered. The intuition is
simple. At a net worth level w
b − ² both types receive zero in the LCSE because a low type, revealed as such,
cannot satisfy his budget constraint. At this point, both types would be better off pooling and achieving
budget balance in expectation (across types).
Proposition 5 states that it is possible to support equilibria in which pooling occurs for low net worth
states while separation occurs for higher net worth states.
Proposition 5. In the dynamic option contract game, it is possible to support a default threshold wd < w.
b
There exists wp > w
b such that for w ∈ [wd , wp ] the payoffs resulting from pooling at an allocation aP (w)
are in the equilibrium set. For w > wp there is no pooling equilibrium.
27
Proof. We begin by establishing the existence of a pooling region. Choose ² arbitrarily small and let w ≡ w−².
b
∗
The RSW payoffs at this point are zero. However, if the types were to pool using d = 0, b∗L (w)
b and kL
(w),
b
they could achieve a strictly positive payoff at some s < 1 while the investor earns zero expected profit based
upon priors. The claimed result then follows from Lemma 4. Next recall that there exists a w ≡ wslack at
which the nonmimicry constraint in Program H is slack. For w ≥ wslack , the solution to Program H yields a
greater payoff for the high type than any pooling allocation that gives the investor weakly positive expected
profits based upon prior beliefs. From Lemma 4 it follows that no pooling equilibrium can exist on this
interval.¥
Although pooling extends the continuation region, the firm must default if net worth is sufficiently low.
In particular, the attempt to pool would be unsuccessful if the investor could not break even at s = 1.
Therefore, we define the default threshold in the dynamic option contract game as wd = w
bP , where
w
bP ≡ − max
b,k
π[ρH (b, k) + ΩH (b, k)] + (1 − π)[ρL (b, k) + ΩL (b, k)] − k − χγb2 /2.
(49)
Based upon Lemma 4, we can use the following Program P to determine whether pooling is possible at
a particular level of net worth. However, it is worth stressing that pooling equilibria need not be unique.
For example, at some points on the net worth space there are pooling equilibria where only equity finance
is employed. Program P makes the high type as well off as possible subject to: (1) The pooling allocation
is improving for the low type relative to his separating allocation (LTI). (2) Provision of outside funding is
profitable in expectation (PIE) based upon the investor’s prior beliefs.
PROGRAM P:
max
a∈A
d + (1 − s)ΩH
subject to
LT I
: d + (1 − s)ΩL ≥ d∗L + (1 − s∗L )ΩLL
P IE
: π[ρH (b, k) + ΩH (b, k)] + (1 − π)[ρL (b, k) + ΩL (b, k)] ≥ k + χγb2 /2 − w.
We denote the solution to Program P as aP ≡ (bP , dP , kP , sP ). From Lemma 4 we know that pooling is only
possible at point w if dP + (1 − sP )ΩH ≥ d∗H + (1 − s∗H )ΩHH .
∗∗
We define equilibrium allocations in the option contract game (a∗∗
L , aH ) point-wise on the net worth
space with
dP (w) + [1 − sP (w)]ΩH
∗∗
P
P
≥ d∗H (w) + [1 − s∗H (w)]ΩHH ⇒ (a∗∗
L , aH )(w) ≡ (a (w), a (w))
dP (w) + [1 − sP (w)]ΩH
∗∗
∗
∗
< d∗H (w) + [1 − s∗H (w)]ΩHH ⇒ (a∗∗
L , aH )(w) ≡ (aL , aH )(w).
28
(51)
The final step in the construction is to use a recursive equation to pin down the equity value functions
vj (w) ≡ πHj
"
d∗∗
H
+ (1 −
+(1 − π Hj )
"
d∗∗
L
s∗∗
H )β
Z
∞
εd
HH
+ (1 −
vH [(1 −
s∗∗
L )β
Z
∞
εd
LL
∗∗
δ)kH
+
vL [(1 −
∗∗ α
θH ε(kH
)
∗∗
δ)kL
+
−
b∗∗
H ]f (ε)dε
∗∗ α
θL ε(kL
)
−
#
b∗∗
L ]f (ε)dε
(52)
#
.
Before turning to the results of the numerical simulation, we present the optimality conditions from
Program P. Letting λ denote the multiplier on the PIE constraint, the optimal financing policy in the
pooling equilibrium satisfies
H
L
L
π(ΩH
b + ρb ) + (1 − π)(Ωb + ρb ) + χγb =
and the optimal investment rule satisfies
π(ΩH
K
+
ρH
K)
+ (1 −
π)(ΩL
K
+
ρL
K)
¯ ¯ L ¯¸
∙¯
¯ ¯ Ωb ¯
(1 − λπ)ΩH (1 − s) ¯¯ ΩH
b ¯
¯
−¯
¯
H
λ
Ω ¯ ¯ ΩL ¯
¸
∙
(1 − λπ)ΩH (1 − s) ΩH
ΩL
k
k
−1=
− L .
λ
ΩH
Ωk
k
(53)
(54)
From the optimality conditions on d and s it is also possible to show that 1 > λπ. The left sides of the two
optimality conditions represent the average deadweight loss associated with increases in debt and capital,
respectively. Since the high type is pooling, he is concerned about the average level of efficiency. The right
side of the optimality conditions capture the cross-subsidy provided by the high type to the low type in any
pooling equilibrium.
Figure 5 presents the results from simulations under the same parameterization as in Section 4, but allows
for the possibility of pooling. Consistent with Proposition 4, the firms pool when net worth is low and switch
to the LCSE when net worth is sufficiently high. On the pooling interval, the high type underinvests and the
low type overinvests relative to first-best. Once there is a switch to the LCSE, the high type sharply increases
investment and the low type sharply cuts investment. Figure 6 depicts equilibrium financing policies. On
the pooling region, investment is financed primarily with debt, since this reduces cross-subsidies from the
high type to the low type. Once again, we see that the switch from pooling to separating equilibrium leads
to non-monotonic policies. For example, the debt of the high type spikes upward once there is a switch from
pooling to separating equilibrium. The debt of the high type then begins falling as increases in net worth
reduce the need for costly signaling in the separating equilibrium.
A unique feature of our structural model is that it differentiates between dividends and share repurchases
as modes of distributing cash to shareholders. From this perspective, it is interesting to note the share
repurchases occur in equilibrium once we allow for pooling. This did not occur when we only considered
LCSE. The intuition is simple. When we allow for pooling, the continuation region is larger and default
costs are lower. In this case, the logic of the model more closely approximates that of Constantinides and
Grundy (1990), where firms issue debt in excess of the amount needed for investment and use the residual
29
funds for share repurchases. In other words, when bankruptcy becomes less costly, the firm places greater
reliance on financial signals (as distinct from real investment signals).
6
Integration with Trade-off Theory
The model can be extended to include a corporate income tax and the associated tax shield benefit of
debt. There are a number of compelling rationales for extending the model to include taxes. First, there
is no reason why theories of financing based upon hidden information and the trade-off theory should be
mutually exclusive. It is noteworthy that this point was stressed by Myers (1984) himself. Second, including
tax advantages of debt will help the model to explain an even broader set of stylized facts. In particular,
Graham (1996) presents empirical evidence in support of the hypothesis that increases in the value of the
debt tax shield result in marginal increases in the propensity of firms to use debt finance.
Following the convention in the literature, we assume the corporate income tax rate is a constant
τ ∈ (0, 1). The base of the corporate income tax is operating income less economic depreciation less interest
expense plus interest income. Let y denote the promised yield to maturity on debt. From the identity
ρ ≡ b/(1 + y), it follows that interest expense is yρ = b − ρ. The same expression represents taxable interest
income if b < 0. In the U.S. payments to lenders are treated as principal first. To capture this rule, we
assume interest deductions are disallowed in the event of default. Finally, we note that the inclusion of a
corporate income tax creates a tax penalty to financial slack since shareholders can earn the rate of return
r by saving on their own account while the corporation earns r(1 − τ ) after-tax. To isolate this tax effect,
below we set agency costs of financial slack equal to zero (γ = 0).
To summarize, this subsection replaces Assumption 3 with Assumption 30 .
Assumption 30 . Corporate income is taxed at rate τ ∈ (0, 1). In the event of default, interest deductions are
disallowed. Default is endogenous. In the event of default, the lender has strict seniority. The deadweight
¯ ¯
default cost is a fraction (φ) of going-concern value ¯wd ¯ . The pre-tax yield on corporate saving is r.
For simplicity, we return to the baseline model that considers only separating equilibria. Accounting for
the corporate income tax, provisional net worth is
w
e ≡ (1 − δ)k + θεk α − b − τ [θεkα − δk − (b − ρ)] = (1 − δ(1 − τ ))k + (1 − τ )θεkα − b + τ (b − ρ).
(55)
From (55) it follows that the sum of net worth and the debt payment is just equal to the internal resources
of an unlevered firm plus the value of the debt tax shield. The default-inducing shock must be modified to
account for the effect of taxes, with
[1 − δ(1 − τ )]kj + (1 − τ )θi εdij kjα − bj + τ (bj − ρj ) ≡ wd ⇒ εdij ≡
30
bj − [1 − δ(1 − τ )]kj + wd − τ (bj − ρj )
. (56)
(1 − τ )θi kjα
Accounting for the fact that interest deductions are disallowed in default, lender recoveries in the event
of default are equal to
¯ ¯
(1 − δ(1 − τ ))ki + (1 − τ )θi εkiα + (1 − φ) ¯wd ¯ .
(57)
That is, in the event of a default, the lender receives the value of the physical capital, earnings, and goingconcern value net of the corporate income tax bill.
The expected end-of-period equity value must also be adjusted to account for the new definition of net
worth
ij
Ω ≡β
Z
∞
εd
ij
vi [(1 − δ(1 − τ ))kj + (1 − τ )θi εkjα − bj + τ (bj − ρj )]f (ε)dε,
(58)
and the bond value function must account for the new definition of lender recoveries in default
" Z
ρ ≡ β bi
i
∞
f (ε)dε +
εd
ii
Z
εd
ii
−∞
[(1 − δ(1 − τ ))ki + (1 −
τ )θi εkiα
d
#
− (1 − φ)w ]f (ε)dε .
(59)
In the interest of brevity, we focus on financial policies. Once again, the low type allocation is derived
by solving Program L. Program L is unaffected save the new definitions for the equity value and debt value
which account for taxes. The optimality condition for b∗L is
∂
(yL ρL ) × τ
∂b
Z
∞
f (ε)dε =
εd
LL
¯ ¯
∂εd
f (εdLL ) LL φ ¯wd ¯
∂b
#
∙
¸ "Z ∞
∂
0
+ 1 − τ (yL ρL )
[vL − 1]f (ε)dε .
∂b
εd
LL
(60)
The left side of equation (60) is the marginal tax shield benefit provided by debt. The first term on the right
side of the equation is the marginal bankruptcy cost. The second term on the right side is a measure of the
0
precautionary cost of debt service. Since asymmetric information induces vL
> 1 for low net worth states,
the low type is predicted to have low debt relative to traditional trade-off theory. Thus, the precautionary
effect offers a potential explanation for what would appear to be conservative debt policies.
The high type allocation again solves Program H. In the interest of brevity, we redefine the marginal cost
of debt service to account for taxes. The marginal effect of b on the expected discounted end-of-period value
of shareholders’ equity for a high-type (low-type) that has taken the high-type allocation is
ΩHH
b
ΩLH
b
∙
¸ Z ∞
∂yρ
0
α
≡ − 1−τ
vH
[(1 − δ(1 − τ ))kH + (1 − τ )θH εkH
− bH + τ (bH − ρH )]f (ε)dε (61)
β
∂b
εd
HH
∙
¸ Z ∞
∂yρ
0
α
≡ − 1−τ
vL
[(1 − δ(1 − τ ))kH + (1 − τ )θL εkH
− bH + τ (bH − ρH )]f (ε)dε.
β
d
∂b
εLH
The optimality condition for b∗H is
¯ ¯
¯¸
¶ ∙¯
¯ ¯ ds
¯
¯ ds
¯
¯
¯
f (ε)dε +
(aH ; θL )¯ − ¯ (aH ; θH )¯¯
¯
db
db
εd
HH
"Z
#
∙
¸
∞
∂
∂εdHH ¯¯ d ¯¯
d
0
= f (εHH )
[vH − 1]f (ε)dε .
φ w + 1 − τ (yH ρH )
∂b
∂b
εd
HH
∂
(yH ρH ) × τ
∂b
Z
∞
µ
μΩLH
βλH
31
(62)
The financing of the high type equates marginal tax and signaling benefits of debt with bankruptcy and
precautionary costs. If net worth is sufficiently high, the nonmimicry constraint is slack and the precautionary
effect induces the high type to have low leverage relative to that predicted by traditional trade-off theory.
This offers another explanation for Graham’s (2000) finding that large-liquid firms have conservative debt
policies.16 By way of contrast, if net worth is low, the signaling motive is operative. This induces high
leverage relative to that predicted by trade-off theory.
7
Conclusions
The objective of this paper was to take a first step in constructing dynamic structural models of corporate
financing when insiders have private information. Beginning with a static model, we showed that firms with
positive information will signal through the issuance of defaultable debt and increase investment in order
to reduce associated default costs. These two effects are also present in our dynamic model. The dynamic
model delivers a number of added insights. First, anticipation of future signaling costs converts a risk-neutral
insider into a pseudo-risk-averse insider. The implied precautionary value of internal funds discourages debt
and encourages capital accumulation relative to what one obtains in a static setting. Second, default costs
are lower in a dynamic setting, since continuation value allows the firm to roll-over prior debt obligations.
Finally, we showed that the nature of equilibrium is contingent on the firm’s net worth. In particular, if
net worth is sufficiently low, the costs of separation are extremely high and firms should find their way to a
Pareto dominating pooling equilibrium.
The most obvious direction to take this line of research is to analyze optimal security design. In this
paper, we took the set of securities as given, and derived equilibria given this set. We anticipate that the
securities emerging as optimal under repeated hidden information will be similar to those that are optimal
under repeated moral hazard. Whether the problem is one of hidden information or hidden action, low
reported values of operating profits must be punished. In our model, low reported profits are punished with
transfers of ownership in default. A more general model, with long-term debt, would allow for intermediate
punishments in the form of higher interest rates.
16 This explanation differs fundamentally from that offered by Hennessy and Whited (2005), which is based upon taxes on
distributions and direct flotation costs. In the present model, we have shut off those two channels.
32
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35
Appendix A: Proofs
Proof of Lemma 1.
The allocation a∗L was in the feasible set for Program H. To verify, the N MLH constraint would be trivial as
the allocations would be type-independent. Since BCL is satisfied at a∗L and s∗L ≥ 0 we know BCH would
be slack. The fact that a∗H solves Program H tells us the high type’s maximand must be at least as high at
a∗H as at a∗L .
Proof of Lemma 3.
First note that the nonmimicry constraint must be binding at wd since the high type gets a zero payoff here
but would achieve a strictly positive payoff if he needed to satisfy only BCH but could ignore N MLH . Thus,
the nonmimicry constraint binds for sufficiently low net worth. Next consider high levels of net worth and
solve Program H ignoring the nonmimicry constraint For w sufficiently high the solution to that program
entails sH = 0 and bH < 0. Thus, the high type is not floating any securities and the low type cannot gain
from mimicry since there is no gain from securities mispricing.
Applying the Envelope Theorem, the value of internal funds if the low type is realized is λL = 1. The
value of a dollar of internal funds to the high type is given by λH + μ. To see this, it must be noted that the
low type’s equilibrium payoff (which enters the N MLH constraint) can be expressed as w + κ∗ . This explains
why the shadow value of internal funds to the high type is not simply λH . Taking a probability weighted
average of these values yields (34).¥
Appendix B: Verification of PBE
We now verify that the solutions to Programs L and H in conjunction with beliefs (9) constitute a PBE.
First note that any offer a0 ∈
/ {a∗L , a∗H } that would be acceptable to the investor must satisfy both BCL
and BCH , because the beliefs minimize the value of the package of securities. We next verify that it is
optimal for insiders observing θL and θH to offer to the investor a∗L and a∗H , respectively. Consider first
the low type’s incentive to offer some allocation a0 ∈
/ {a∗L , a∗H } that would be acceptable to the investor.
Since a0 is acceptable, it must satisfy BCL . Thus, a0 was in the feasible set for Program L and optimality
implies the low type prefers a∗L to a0 . This same argument shows the low type will not make an offer that is
rejected, since such an offer induces zero outside funding, and zero outside funding is always acceptable to
the investor. Therefore, the low type prefers a∗L to all offers other than a∗H . Finally, N MLH ensures the low
type prefers a∗L to a∗H .
Consider next the high type’s incentive to offer some allocation a0 6= a∗H that would be accepted by the
investor. Note that the allocation a0 was in the feasible set for Program H. To see this, note that BCH is
necessarily satisfied, since the offer is acceptable even with worst-case beliefs. Since a0 also satisfies BCL ,
we know a0 was in the feasible set for Program L. The optimality of a∗L in Program L implies that N MLH
would be satisfied if the high type were to offer a0 . Thus, a0 was in the feasible set for Program H, and the
36
optimality of a∗H for this program implies the high type prefers a∗H to a0 . This same argument shows that
the high type will not make an offer that is rejected, since such an allocation is equivalent to zero outside
funding, and zero outside funding is always acceptable to the investor.
Appendix C: Optimality Conditions in Dynamic Model
The Lagrangian for Program L is
L = dL + (1 − sL )ΩLL + λL [w − dL − kL − χγb2L /2 + sL ΩLL + ρL ] + ηL dL + ψ L (1 − sL ).
The first-order conditions for dL and sL are
1 − λL + η L = 0
(63)
(λL − 1)ΩLL − ψ L = 0.
(64)
From equations (63) and (64) it follows that η L ΩLL = ψ L . Since ε has unbounded support, ψ L (w) > 0
cannot occur for w > wd . To see this, note that the proposed equilibrium would entail the low type getting
zero. But then the low type would always opt for the high type allocation unless sH = 1 and dH = 0. Of
course, this implies both types get a payoff of zero which contradicts being on shareholders’ continuation
region.
The first-order condition pinning down b∗L is
β
"Z
∞
εd
LL
0
[vL
((1
− δ)kL +
α
θL εkL
#
− bL ) − 1]f (ε)dε = −χγbL + β
∂εdLL
d
f (εdLL )φwL
.
∂bL
(65)
From Lemma 2 it follows that the left side of (65) is positive. It follows that bL must be negative. The
optimality condition for kL takes a similar form, with
β
∙Z
ε
∞
0
[vL
((1
− δ)kL +
α
θL εkL
− bL )][1 − δ +
¸
α−1
αθL εkL
]f (ε)dε
= 1.
The Lagrangian for Program H is
L = dH + (1 − sH )ΩHH + λH [w − dH − kH − χγb2H /2 + sH ΩHH + ρH ]
+μ[d∗L + (1 − s∗L )ΩLL − dH − (1 − sH )ΩLH ] + ηH dH + ψ H (1 − sH ).
The first-order conditions for dH and sH are
1 − λH − μ + ηH = 0
37
(66)
(λH − 1)ΩHH + μΩLH − ψ H = 0
(67)
Substituting (66) into (67), we obtain
ηH ΩHH = μ(ΩHH − ΩLH ) + ψ H .
(68)
It is straightforward to establish ψ H (w) = 0 on the continuation region (w > wd ). To see this, suppose to
the contrary that ψ H > 0. It follows from (68) that η H > 0 and dH = 0. Thus, the high type gets a payoff
of zero. However, the low type would then also receive an allocation with a payoff of zero since the N MHL
constraint demands
dH + (1 − sH )ΩHH ≥ dL + (1 − sL )ΩHL ≥ dL + (1 − sL )ΩLL .
(69)
But this contradicts being on shareholders’ continuation region. Without loss of generality we shall treat
ψ H = 0 as we solve for optimal policies on the continuation region.
Rearranging (68) we obtain
η H = μ[(ΩHH − ΩLH )/ΩHH ].
(70)
Condition (70) tells us that whenever N MLH binds, the dividend is zero.
Appendix D: Details of the Numerical Algorithm
The solution procedure is based on value function iteration. The individual steps are as follows. The
idiosyncratic shock ε is implemented by discretizing its domain using N possible values. Each maximization
is implemented by discretizing the domain of the decision variables.
1. Guess going-concern value wd .
2. Guess the end-of-period equity value functions (vj ) which are vectors on the net worth space.
3. For each point in the net worth grid, find the allocation aL that maximizes the objective function of
the low type subject to its budget constraint. Since the dividend is not unique, pick the allocation in the
optimal set that minimizes the dividend.
4. For each point in the net worth grid, find the allocation aH that maximizes the high type’s objective
subject to the budget and nonmimicry constraints.
0
5. Using the solutions from steps 3 and 4, compute new value functions vj using the recursive equation
0
"
vj = πHj dH + β(1 − sH )
"
+ π Lj dL + β(1 − sL )
N
X
n=1
N
X
n=1
f (εn )vH [(1 − δ)kH +
α
θH εn kH
− bH ]
#
α
f (εn )vL [(1 − δ)kL + θL εn kL
− bL ] .
38
#
0
6. The functions vj from the previous step are the new guesses for vj . The procedure is then restarted
from step 2 until convergence.
7. Check that wd satisfies the endogenous default condition vj (wd ) = 0. If these conditions are not
satisfied, update the initial guess wd and restart the procedure from step 1 until convergence.
39
Figure 1: Single-crossing conditions
This figure presents single-crossing conditions. Panel A shows the indifference curves drawn under the assumption that the
low type is more willing to exchange equity for a reduction in debt. Therefore, debt provides a positive signal and the high
type borrows more/saves less in order to separate from the low type. Panel B shows the indifference curves drawn under the
assumption that the low type has a greater willingness to exchange equity for capital. In this case, higher capital investment
provides a negative signal which encourages underinvestment relative to first-best.
Panel A: Debt-for-Equity Substitution as Positive Signal
s
6
u(b, d, k, s; θL ) = const.
sH
u(b, d, k, s; θH ) = const.
bH
b
Panel B: Equity-Financed Investment as Negative Signal
s
6
u(b, d, k, s; θL ) = const.
u(b, d, k, s; θH ) = const.
sH
k
H
k
40
Figure 2: Enterprise value of the firm
The enterprise value of the firm v − w is plotted as a function of the realized net worth, w. The productivity shock ε is
exponentially distributed and both productivity shocks are i.i.d.
3.36
3.35
Enterprise Value
3.34
3.33
3.32
3.31
3.3
3.29
−4
−2
0
2
4
w
41
6
8
10
Figure 3: Equilibrium capital allocations
Equilibrium capital allocations, ki∗ , scaled by the first-best allocations, kiF B , are plotted as a function of the realized net worth,
w, for both high and low values of θ. The productivity shock ε is exponentially distributed and both productivity shocks are
i.i.d.
1.16
H
L
1.14
1.12
k*/kFB
1.1
1.08
1.06
1.04
1.02
1
−4
−2
0
2
4
w
42
6
8
10
Figure 4: Equilibrium financing policies
Equilibrium financing policies: debt, ρi , and equity, s∗i Ωii , as well as equilibrium dividend policy, d∗i , are plotted as functions
of the realized net worth, w. Panel A presents the case of high value of θ, while Panel B presents the case of low value of θ.
The productivity shock ε is exponentially distributed and both productivity shocks are i.i.d.
Panel A: High value of θ
8
7
Debt
Total Issued Equity
Dividends
6
5
Value
4
3
2
1
0
−1
−4
−2
0
2
4
6
8
10
6
8
10
w
Panel B: Low value of θ
9
8
Debt
Total Issued Equity
Dividends
7
6
Value
5
4
3
2
1
0
−1
−4
−2
0
2
4
w
43
Figure 5: Equilibrium capital allocations in option contract game
Equilibrium capital allocations, ki∗ , scaled by the first-best allocations, kiF B , are plotted as a function of the realized net worth,
w, for both high and low values of θ. The productivity shock ε is exponentially distributed and both productivity shocks are
i.i.d.
1.5
H
L
1.4
1.3
k*/kFB
1.2
1.1
1
0.9
0.8
0.7
−4
−2
0
2
4
w
44
6
8
10
Figure 6: Equilibrium financing policies in option contract game
Equilibrium financing policies: debt, ρi , and equity, s∗i Ωii , as well as equilibrium dividend policy, d∗i , are plotted as functions
of the realized net worth, w. Panel A presents the case of high value of θ, while Panel B presents the case of low value of θ.
The productivity shock ε is exponentially distributed and both productivity shocks are i.i.d.
Panel A: High value of θ
7
Debt
Total Issued Equity
Dividends
6
5
4
Value
3
2
1
0
−1
−2
−4
−2
0
2
4
6
8
10
6
8
10
w
Panel B: Low value of θ
9
8
Debt
Total Issued Equity
Dividends
7
6
Value
5
4
3
2
1
0
−1
−4
−2
0
2
4
w
45
Table 1: Leverage Regressions
ρ
This table reports results of several regressions on the simulated data. The first three regressions have book leverage, kt , as
t
ρ
ρ
t −wt
the dependent variable. The last regression has kt − kt−1 as the dependent variable. The financing gap is defined as dt +k
k(t)
t
t−1
and operating profits are defined as θ t εt ktα−1 . The simulated panel of firms contains 3,000 firms over 31 time periods, where
the initial period has been dropped for each firm. The productivity shock ε is exponentially distributed and both productivity
shocks are i.i.d.
E[ kρ ] −
ρt−1
kt−1
Financing Gap
Lagged Operating Profits
Book-to-Market
0.2405
( 0.0912 )
-
-
-
-
-0.0114
( 0.0044 )
-
-
-
-0.0075
( 0.0067 )
0.0103
( 0.0090 )
-
-
-
-
0.1813
( 0.0194 )
Table 2: Announcement Effect Regressions
This table reports results of several regressions on the simulated data with the abnormal return on the announcement day,
d +(1−s )Ωii −v(w )
t
it
t
ARt = it
, as the dependent variable. The investment rate is defined as kit . Notation a± means conditioning
v(wt )
t
on the positive(negative) values of a. The simulated panel of firms contains 3,000 firms over 31 time periods, where the initial
period has been dropped for each firm. The productivity shock ε is exponentially distributed and both productivity shocks are
i.i.d.
ρ+
t
ρt
kt
st Ωt
kt
-
-
-
-
0.0752
( 0.0251 )
-
-
0.0155
( 0.0017 )
0.0103
( 0.0094 )
-
-
-
-
0.1307
( 0.0595 )
-
0.0162
( 0.0052 )
-
0.0222
( 0.0512 )
-0.0042
( 0.0277 )
Investment Rate
0.0163
( 0.0012 )
+
ρ+
t +st Ωt
46